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A mathematical axiom cannot be wrong, since by definition an axiom is always true. More accurately, a system of axioms can be vacuous, in the sense that it lea

A mathematical axiom cannot be wrong, since by definition an axiom is always true. More accurately, a system of axioms can be vacuous, in the sense that it leads to conclusions that are logically impossible. To ask "Is the axiom of choice true" requires that we agree on the meaning of 'truth', which is of course a very complicated philosophical question. The author tries to make an appeal to our intuition by describing an example that (supposedly) takes place in the real world, but the real world is the realm of physics. In physics, truth derives from the result of experiments; whereas in mathematics, true statements are derived from a set of axioms. Considering that an axiom is a rule that is made up by a person, and that there are an infinite number of consistent axiom systems, the question of the "truth" of a set of axioms has no meaning, unless we are actually asking, is a given axiom necessary for an accurate description of nature. As far as I am aware there is no part of physics in which the axiom of choice plays any direct role.

j2kun 12y ago
A lot of people who talk about math on the internet seem to phrase it as this cold austere thing where it doesn't matter what anyone thinks because theorems are theorems and it's all just axioms in the end. And while we all sort of work with that understanding (somewhere, deep down there), in our waking hours "experiments" are the most efficient and insightful way to check our work and intuition. If we come up with some great beautiful theory and, assuming all the work was done correctly, we find a simple example for which our results contradict something we know to be true (or hold dear) then we have to look more closely at the starting assumptions, potentially rejecting them entirely. These kinds of mathematical experiments definitely do not "take place" in the real world. Having an infinite number of people who can see infinitely far in front of them should have been a clue to that. It's just a way to expand mathematical jargon into an idea that is easy to keep in your head.

j2kun 12y ago

A lot of people who talk about math on the internet seem to phrase it as this cold austere thing where it doesn't matter what anyone thinks because theorems are theorems and it's all just axioms in the end. And while we all sort of work with that understanding (somewhere, deep down there), in our waking hours "experiments" are the most efficient and insightful way to check our work and intuition. If we come up with some great beautiful theory and, assuming all the work was done correctly, we find a simple example for which our results contradict something we know to be true (or hold dear) then we have to look more closely at the starting assumptions, potentially rejecting them entirely. These kinds of mathematical experiments definitely do not "take place" in the real world. Having an infinite number of people who can see infinitely far in front of them should have been a clue to that. It's just a way to expand mathematical jargon into an idea that is easy to keep in your head.